1549380232-Automorphic_Forms_and_Applications__Sarnak_

154 J.W. COGDELL, £-FUNCTIONS FOR GLn

would like to have simply that Ft.Un) = 0. Let u = (u1, ... , Un-1) E A_n-l and
consider the function

ft,(u) =Ft. cn-1 ~).

Now ft,(u) is a function on kn-l\A_n-l and as such has a Fourier expansion

ft,(u) = L ]e,(a)'l/Ja(u)

aEkn-1

where 'l/Ja(u) = 'l/;(a .tu) and

fe,(a) = r fc.(u)'l/J-a(u) du.

}kn- 1\A_n- 1

In this language, the statement lP'~_ 2 Ft.Un-i) = 0 becomes ]e,(en-l) = 0, where as

always, ek is the standard unit vector with O's in all places except the kth where

there is a 1.

Note that Ft,(g) = Ue,(g)-Ve,(g) is left invariant under (Z(k) P(k)) n (Z(k) Q(k))

where Q = Qn_ 2. This contains the subgroup

R(k) = r = a' D!n-1

{ (

In-2

Using this invariance of Fe,, it is now elementary to compute that, with this notation,

Jrr(r)e,(en-1) = }e,(a) where a= (a', D!n-1) E kn-l. Since }e,(en-1) = 0 for all~'

and in particular for II(r)~, we see that for every~ we have ]e,(a) = 0 whenever

D!n-1 -I-0. Thus

fe,(u) = L }t,(a)'l/Ja(u) = L }e,(a', O)'l/J(a',o)(u).

e>Ekn-1 a'Ekn-2

Hence fe,(O, ... , O,un-1) = :Za'Ekn- 2]t,(a^1 ,0) is constant as a function of Un-1·

Moreover, this constant is fe,(O) =Ft.Un), which we want to be 0. This is what our

local condition will guarantee.

If v is a finite place of k, let Ov denote the ring of integers of kv, and let Pv

denote the prime ideal of Ov. We may assume that we have chosen v so that the

local additive character 'l/Jv is normalized, i.e., that 'l/Jv is trivial on Ov and non-trivial

on p;-^1. Given an integer nv ?-: 1 we consider the open compact group

Koo,v(P~v) = {g = (gi,j) E GLn(ov) l(i) 9i,n- l E p~v for 1 :Si :Sn -2;

(ii) 9n,j E p~v for 1 :S j :S n - 2;

(iii) 9n,n-l E j)~nv}.

(As usual, 9i,j represents the entry of gin the i-th row and j-th column.)

Lemma 5.1. Let v be a finite place of k as above and let (IIv, VrrJ be an irreducible

admissible generic representation of GLn(kv)· Then there is a vector~~ E Vrrv and

a non-negative integer n v such that

(1) for any g E Koo,v(!J~v) we have IIv(g)~~ = wrrJ9n,n)~~

J (

In-2 )

(2) IIv 1 u ~~ du = 0.

Pv -1^1